Department of Management Sciences, Pune University (PUMBA)

Yes, they do accept MAT for MBA as you have scored 98.11 percentile. You are one edge upper than average and chances of selection are higher. So proceed and apply for the further steps of admission process.
Best colleges for management amongst top 50 you can check for:
1) Great Lakes Chennai
2) Rajagiri Centre for Business Studies, Kochi (Scholarship available for Meritorious Students)
3) Christ University, Bangalore
4) LIBM, Chennai
5) Tezpur University (Central University)

To get admission into PUMBA, you need to have percentile above 97 irrespective of any examination you have appeared for. You have got 94 percentile in MAT, so all I can say just wait for the cut off to break down. For getting admission into PUMBA currently your scenario is 50-50.

Yes, OMS candidates are very much welcomed in PUMBA. You have 98.11 percentile in MAT which is really a very good score and yes you can keep high hopes for getting admission into PUMBA.

For a non-Maharashtra students, it is very rare that one will get admission through CAT score, because the DTE consider all the exams like GMAT, ATMA at the same level and cut off for which is above 98%. So, mostly the students with percentile in MAT and ATMA will get the admission above 98 percentile.

Yes, MBA+ offers dual specialization one of them is minor and other is major.

PUMBA does not offer any PGDM course, there are no management seats, all the admissions are done on the basis of the final merit list.

MAT score is not accepted for admission in any of the courses in PUMBA. Score of exams like MAT, CMAT and CAT is accepted for MBA+ and for MBA (PHARMA-BT) score of ATMA is accepted.

There is no management quota in PUMBA. PUMBA is a very reputed MBA institute. So, in PUMBA there is no chance to get admission through backdoor entry.

Yes, you are 100% getting a call for admission. SIESCOMS is better than PUMBA in terms of education. After that Great Lakes-PGDM, XIME, SIES, ISBM-Pune, etc., can also be preferred.